Polynomial-like iterative equation on Riesz spaces

In this paper we investigate the polynomial-like iterative equation on Riesz spaces. Since a Riesz space does not need to have a metric space structure, neither the Schauder fixed point theorem nor the Banach fixed point theorem is available. Using the Knaster–Tarski fixed point theorem, we first obtain the existence and uniqueness of order-preserving solutions on convex complete sublattices of Riesz spaces. Then, restricting to \(\mathbb {R}\) and \(\mathbb {R}^n\), special cases of Riesz space, we obtain semi-continuous solutions and integrable solutions, respectively. Finally, we present more special cases of Riesz space in which solutions to the iterative equation can be discussed.